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Rademacher Complexity
In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of real-valued functions with respect to a probability distribution. Definitions Rademacher complexity of a set Given a set A\subseteq \mathbb^m, the Rademacher complexity of ''A'' is defined as follows:Chapter 26 in : \operatorname(A) := \frac \mathbb_\sigma \left \sup_ \sum_^m \sigma_i a_i \right where \sigma_1, \sigma_2, \dots, \sigma_m are independent random variables drawn from the Rademacher distribution i.e. \Pr(\sigma_i = +1) = \Pr(\sigma_i = -1) = 1/2 for i=1,2,\dots,m, and a=(a_1, \ldots, a_m). Some authors take the absolute value of the sum before taking the supremum, but if A is symmetric this makes no difference. Rademacher complexity of a function class Let S=(z_1, z_2, \dots, z_m) \in Z^m be a sample of points and consider a function class \mathcal of real-valued functions over Z^m. Then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Learning Theory
In computer science, computational learning theory (or just learning theory) is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms. Overview Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples, including samples that have not been seen previously by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made on new samples. In addition to performance bounds, computational learning theory studies the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz Constant
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (or '' modulus of uniform continuity''). For instance, every function that has bounded first derivatives is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of strict incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent And Identically Distributed Random Variables
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as ''i.i.d.'', ''iid'', or ''IID''. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covering Number
In mathematics, a covering number is the number of spherical balls of a given size needed to completely cover a given space, with possible overlaps. Two related concepts are the ''packing number'', the number of disjoint balls that fit in a space, and the ''metric entropy'', the number of points that fit in a space when constrained to lie at some fixed minimum distance apart. Definition Let (''M'', ''d'') be a metric space, let ''K'' be a subset of ''M'', and let ''r'' be a positive real number. Let ''B''''r''(''x'') denote the ball of radius ''r'' centered at ''x''. A subset ''C'' of ''M'' is an ''r-external covering'' of ''K'' if: :K \subseteq \bigcup_ B_r(x). In other words, for every y\in K there exists x\in C such that d(x,y)\leq r. If furthermore ''C'' is a subset of ''K'', then it is an ''r-internal covering''. The external covering number of ''K'', denoted N^_r(K), is the minimum cardinality of any external covering of ''K''. The internal covering number, denoted N^_r( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Ball
Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (album), 1997 album by the Australian band Regurgitator * The Units, a synthpunk band Television * '' The Unit'', an American television series * '' The Unit: Idol Rebooting Project'', South Korean reality TV survival show Business * Stock keeping unit, a discrete inventory management construct * Strategic business unit, a profit center which focuses on product offering and market segment * Unit of account, a monetary unit of measurement * Unit coin, a small coin or medallion (usually military), bearing an organization's insignia or emblem * Work unit, the name given to a place of employment in the People's Republic of China Science and technology Science and medicine * Unit, a vessel or section of a chemical plant * Blood unit, a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vapnik–Chervonenkis Dimension
Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the capacity (complexity, expressive power, richness, or flexibility) of a set of functions that can be learned by a statistical binary classification algorithm. It is defined as the cardinality of the largest set of points that the algorithm can shatter, which means the algorithm can always learn a perfect classifier for any labeling of at least one configuration of those data points. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so it can fit a given set of training points well. But one can expect that the classifier will make errors on other points, because it is too wigg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dudley's Theorem
In probability theory, Dudley's theorem is a result relating the expected upper bound and regularity properties of a Gaussian process to its entropy and covariance structure. History The result was first stated and proved by V. N. Sudakov, as pointed out in a paper by Richard M. Dudley. Dudley had earlier credited Volker Strassen Volker Strassen (born April 29, 1936) is a German mathematician, a professor emeritus in the department of mathematics and statistics at the University of Konstanz. For important contributions to the analysis of algorithms he has received many aw ... with making the connection between entropy and regularity. Statement Let (''X''''t'')''t''∈''T'' be a Gaussian process and let ''d''''X'' be the pseudometric on ''T'' defined by :d_(s, t) = \sqrt. \, For ''ε'' > 0, denote by ''N''(''T'', ''d''''X''; ''ε'') the entropy number, i.e. the minimal number of (open) ''d''''X''-balls of radius ''ε'' required to cover ''T''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Growth Function
The growth function, also called the shatter coefficient or the shattering number, measures the richness of a set family. It is especially used in the context of statistical learning theory, where it measures the complexity of a hypothesis class. The term 'growth function' was coined by Vapnik and Chervonenkis in their 1968 paper, where they also proved many of its properties. It is a basic concept in machine learning., especially Section 3.2 Definitions Set-family definition Let H be a set family (a set of sets) and C a set. Their ''intersection'' is defined as the following set-family: : H\cap C := \ The ''intersection-size'' (also called the ''index'') of H with respect to C is , H\cap C, . If a set C_m has m elements then the index is at most 2^m. If the index is exactly 2''m'' then the set C is said to be shattered by H, because H\cap C contains all the subsets of C, i.e.: : , H\cap C, =2^, The growth function measures the size of H\cap C as a function of , C, . For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
VC Dimension
VC may refer to: Military decorations * Victoria Cross, a military decoration awarded by the United Kingdom and also by certain Commonwealth nations ** Victoria Cross for Australia ** Victoria Cross (Canada) ** Victoria Cross for New Zealand * Victorious Cross, Idi Amin's self-bestowed military decoration Organisations * Ocean Airlines (IATA airline designator 2003-2008), Italian cargo airline * Voyageur Airways (IATA airline designator since 1968), Canadian charter airline * Visual Communications, an Asian-Pacific-American media arts organization in Los Angeles, US * Viet Cong (also Victor Charlie or Vietnamese Communists), a political and military organization from the Vietnam War (1959–1975) Education * Vanier College, Canada * Vassar College, US * Velez College, Philippines * Virginia College, US Places * Saint Vincent and the Grenadines (ISO country code), a state in the Caribbean * Sri Lanka (ICAO airport prefix code) * Watsonian vice-counties, subdivisions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Family
In set theory and related branches of mathematics, a collection F of subsets of a given Set (mathematics), set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. The term "collection" is used here because, in some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a Class (set theory), proper class rather than a set. A finite family of subsets of a finite set S is also called a ''hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the power set of S and is denoted by \wp(S). The power set \wp(S) of a given set S is a family of sets over S. A subset of S having k elements is called a Finite set#Definition and terminology, k-subset of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
